To test the scaling properties of Alg. REF , we apply it on a slightly modified variant of the hanging chain example — a popular benchmark for numerical methods for NMPC [1]}, [2]}. Parameter values used in the following description are given in Table REF . The system is defined by a chain of masses \(m\)  connected by linear springs with stiffness \(k\) . The mass at one end of the chain is tied to the origin; the \(n_u = 3\) velocity components of the other end's mass constitute the control inputs of the system. The system state is given by the position components of the controlled mass as well as the position and velocity components of the intermediate masses, resulting in a state space dimension of \(n_x = 6(n_\text{mass} - 2) + 3\). The initial state of the chain is computed based on its resting position where the controlled end is placed at \((x_{\text{init}},y_{\text{init}},z_{\text{init}}) := (6l (n_\text{mass} - 1), 0, 0)\). After applying a control input of \(u_{\text{init}} := (1, 1, 1)\) for \(T_{\text{init}} := 1{s}\), the control task is to restore the chain's resting position while ensuring that none of the masses violates a wall constraint at \(y_{\text{wall}}\) . The example is modified by adding a latent force
\(f_\text{lat}(x,v_x) := \alpha _{\text{lat}} \left(v_x - \sin \left( \beta _1 \frac{2 \pi x}{l} \right) - \sin \left( \beta _2 \frac{2 \pi x}{l} \right)^2 \right)^2\) 
